let a be Real; :: thesis: for S being non empty compact TopSpace
for T being NormedLinearTopSpace
for F, G being Point of (R_NormSpace_of_ContinuousFunctions (S,T))
for f, g being Function of S,T st f = F & g = G holds
( G = a * F iff for x being Element of S holds g . x = a * (f . x) )
let S be non empty compact TopSpace; :: thesis: for T being NormedLinearTopSpace
for F, G being Point of (R_NormSpace_of_ContinuousFunctions (S,T))
for f, g being Function of S,T st f = F & g = G holds
( G = a * F iff for x being Element of S holds g . x = a * (f . x) )
let T be NormedLinearTopSpace; :: thesis: for F, G being Point of (R_NormSpace_of_ContinuousFunctions (S,T))
for f, g being Function of S,T st f = F & g = G holds
( G = a * F iff for x being Element of S holds g . x = a * (f . x) )
let F, G be Point of (R_NormSpace_of_ContinuousFunctions (S,T)); :: thesis: for f, g being Function of S,T st f = F & g = G holds
( G = a * F iff for x being Element of S holds g . x = a * (f . x) )
let f, g be Function of S,T; :: thesis: ( f = F & g = G implies ( G = a * F iff for x being Element of S holds g . x = a * (f . x) ) )
reconsider f1 = F, g1 = G as VECTOR of (R_VectorSpace_of_ContinuousFunctions (S,T)) ;
( G = a * F iff g1 = a * f1 ) ;
hence ( f = F & g = G implies ( G = a * F iff for x being Element of S holds g . x = a * (f . x) ) ) by Th8; :: thesis: verum